Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors

By Gil Kalai

Andrei Raigorodskii

(This post follows an email by Aicke Hinrichs.)

In a previous post we discussed the following problem:

Problem: Let $A$ be a measurable subset of the $d$ -dimensional sphere $S^d = \{x\in {\bf R}^{d+1}:\|x\|=1\}$ . Suppose that $A$ does not contain two orthogonal vectors. How large can the $d$ -dimensional volume of $A$ be?

Setting the volume of the sphere to be 1, the Frankl-Wilson theorem gives a lower bound (for large $d$ ) of $1.203^{-d}$ ,
2) The double cap conjecture would give a lower bound (for large $d$ ) of $1.414^{-d}$ .

A result of A. M. Raigorodskii from 1999 gives a better bound of $1.225^{-d}$ . (This has led to an improvement concerning the dimensions where a counterexample for Borsuk’s conjecture exists; we will come back to that.) Raigorodskii’s method supports the hope that by considering clever configurations of points instead of just $\pm 1$ -vectors and applying the polynomial method (the method of proof we described for the Frankl-Wilson theorem) we may get closer to and perhaps even prove the double-cap conjecture.

What Raigorodskii did was to prove a Frankl-Wilson type result for vectors with $0,\pm1$ coordinates with a prescribed number of zeros. Here is the paper.

Now, how can we beat the $1.225^{-d}$ record???

This entry was posted on July 2, 2009 at 11:05 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors”

Gil Kalai Says:
July 3, 2009 at 3:15 pm
In a later paper, Raigorodskii improves the Borsuk bound
still further, but the “forbidden” scalar products are different
from 0 so it does not seem to have baring on the problem of avoiding pairs of orthogonal vectors.

Combinatorics and more

Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors

One Response to “Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors”

Leave a Reply